Problem 43 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. cost of Tuition The cost of tuition is directly proportional to the number of credits taken. If 11 credits cost \(\$ 720.50,\) find the cost of taking 16 credits. What is the constant of variation?

Step-by-Step Solution

Answer

The constant of variation is \( 65.5 \) and the cost for 16 credits is \( \$1048 \).

Step 1: Understand the Relationship

The problem states that the cost of tuition is directly proportional to the number of credits. This means that the cost can be modeled by the equation \( C = k \times n \), where \( C \) is the cost, \( n \) is the number of credits, and \( k \) is the constant of variation.

Step 2: Find the Constant of Variation

Given that 11 credits cost \( \$720.50 \), we substitute these values into the equation:\[ 720.50 = k \times 11 \]To find \( k \), divide both sides by 11:\[ k = \frac{720.50}{11} = 65.5 \]

Step 3: Calculate the Cost for 16 Credits

Now that we know the constant of variation \( k = 65.5 \), use it to find the cost of 16 credits:\[ C = 65.5 \times 16 \]Calculate the product:\[ C = 1048 \]

Step 4: Conclusion

The cost of 16 credits is \( \$1048 \), and the constant of variation \( k \) is \( 65.5 \).

Key Concepts

Cost of Tuition

The cost of tuition in this problem is directly tied to the number of credit hours a student takes. This type of relationship is known as direct proportionality, where one quantity increases or decreases in tandem with another. In simple terms, as more credit hours are taken, the tuition cost increases proportionally. This relationship can be understood with the equation \( C = k \times n \), where \( C \) represents the total cost of tuition, \( n \) is the number of credits, and \( k \) is the constant of variation.

In the given exercise, if 11 credits cost \( \$720.50 \), it follows that taking more credits will proportionally increase the cost. By setting up the equation correctly, students can easily calculate tuition costs for any number of credits. Understanding this concept allows for efficient tuition budget planning.

Constant of Variation

The constant of variation, denoted as \( k \), serves as an essential part of understanding proportional relationships. It helps quantify how much the tuition cost changes for each credit hour. In other words, it tells us the specific rate at which tuition costs rise with each additional credit.

To find \( k \) in this exercise, the equation \( 720.50 = k \times 11 \) is used. By solving for \( k \), or \( k = \frac{720.50}{11} \), we find that \( k = 65.5 \). This means that each credit hour costs \( \$65.50 \), regardless of the total number of credits taken.

Understanding this constant allows students to predict tutoring costs effectively and gives a clear picture of how tuition scales with increased coursework.

Credit Hours Calculation

Calculating the cost for various credit hours involves applying the concept of direct proportionality using the constant of variation. Once the constant \( k \) is known, it simplifies the process of finding the tuition cost for any number of credit hours.

To calculate the cost for taking 16 credits using \( k = 65.5 \), just plug the values into our fundamental equation: \( C = 65.5 \times 16 \). This calculation results in \( C = 1048 \). Hence, taking 16 credit hours would cost \( \$1048 \).

Understanding how to calculate tuition costs like this can be extremely helpful for students planning their academic journey. It offers a straightforward approach to forecasting educational expenses and ensuring that students can plan financially.